A fuzzy characterization of QF rings

Wenxi Li,Jianlong Chen,Liang Shen

doi:10.1186/1029-242x-2014-184

Wenxi Li, Jianlong Chen + Show 1 more

Open Access

https://doi.org/10.1186/1029-242x-2014-184

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Abstract

Abstract Let R be a ring. R is called a quasi-Frobenius (QF) ring if R is right artinian and R R is an injective right R-module. In this article, we introduce (weak) fuzzy homomorphisms of modules to obtain a fuzzy characterization of QF rings. We also obtain some fuzzy characterizations of right artinian rings and right CF rings. These results throw new light on the research of QF rings and the related CF conjecture. MSC:03E72, 16L60.

Highlights

Recall that a fuzzy subset of a nonempty set X is a map f from X into the closed interval [, ]
The notion of fuzzy subset of a set was firstly introduced by Zadeh [ ]
A ring R is QF if and only if every right R-module can be embedded into a free right R-module

Summary

Introduction

Recall that a fuzzy subset of a nonempty set X is a map f from X into the closed interval [ , ]. The notion of fuzzy subset of a set was firstly introduced by Zadeh [ ] This important ideal has been applied to various algebraic structures such as groups and rings and so on (see [ – ] etc.). We introduce some special fuzzy subsets of modules to characterize quasi-Frobenius (QF) rings. In Section , we use weak fuzzy homomorphisms to give a characterization of injective right R-modules. We obtain some new fuzzy characterizations of right artinian rings. If f satisfies ( ), ( ), ( ), and ( ) of the above conditions, f is called a weak fuzzy homomorphism from MR to NR. A weak fuzzy homomorphism f ∈ WFHomR(M, N) is said to be extendable if there exists g ∈ FHomR(M, N) such that f ≤ g.

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